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  • 1.
    Avelin, Benny
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Lundström, Niklas L.P.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Nyström, Kaj
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Boundary estimates for solutions to operators of p-Laplace type with lower order terms2011Inngår i: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 250, nr 1, s. 264-291Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    In this paper we study the boundary behavior of solutions to equations of the form∇⋅A(x,∇u)+B(x,∇u)=0, in a domain ΩRn, assuming that Ω is a δ-Reifenberg flat domain for δ sufficiently small. The function A is assumed to be of p-Laplace character. Concerning B, we assume that |∇ηB(x,η)|⩽c|η|p−2, |B(x,η)|⩽c|η|p−1, for some constant c, and that B(x,η)=|η|p−1B(x,η/|η|), whenever xRn, ηRn∖{0}. In particular, we generalize the results proved in J. Lewis et al. (2008) [12] concerning the equation ∇⋅A(x,∇u)=0, to equations including lower order terms.

  • 2.
    Avelin, Benny
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Lundström, Niklas L.P.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Nyström, Kaj
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Optimal doubling, reifenberg flatness and operators of p-laplace type2011Inngår i: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 74, nr 17, s. 5943-5955Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    In this paper, we consider equations of p-Laplace type of the form A(x,u)=0. Concerning A we assume, for p∈(1,) fixed, an appropriate ellipticity type condition, Hölder continuity in x and that A(x,η)=|η|p−1A(x,η/|η|) whenever xRn and ηRn∖{0}. Let ΩRn be a bounded domain, let D be a compact subset of Ω. We say that is the A-capacitary function for D in Ω if on D, on Ω in the sense of and in ΩD in the weak sense. We extend to RnΩ by putting on RnΩ. Then there exists a unique finite positive Borel measure on Rn, with support in Ω, such that In this paper, we prove that if Ω is Reifenberg flat with vanishing constant, then for every τ, 0<τ≤1. In particular, we prove that is an asymptotically optimal doubling measure on Ω.

  • 3. Cinti, Chiara
    et al.
    Nyström, Kaj
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Polidoro, Sergio
    A boundary estimate for non-negative solutions to Kolmogorov operators in non-divergence form2012Inngår i: Annali di Matematica Pura ed Applicata, ISSN 0373-3114, E-ISSN 1618-1891, Vol. 191, nr 1, s. 1-23Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We consider non-negative solutions to a class of second-order degenerate Kolmogorov operators of the form L=Sigma(m)(i,j=1)a(i,j)(z)partial derivative(xixj)+Sigma(m)(i=1)a(i)(z)partial derivative(xi)+Sigma(N)(i,j=1)b(i,j)x(i)partial derivative(xj)-partial derivative(t), where z = (x, t) belongs to an open set Omega subset of R-N x R, and 1 <= m <= N. Let (z) over bar is an element of Omega, let K be a compact subset of (Omega) over bar, and let Sigma subset of partial derivative Omega be such that K boolean AND partial derivative Omega subset of Sigma. give sufficient geometric conditions for the validity of the following Carleson type estimate. There exists a positive constant C-K, depending only on Omega, Sigma, K, (z) over tilde and on L, such that sup(K) u <= C(K)u((z) over tilde), for every non-negative solution u of Lu = 0 in Omega such that u|(Sigma) = 0.

  • 4. Cinti, Chiara
    et al.
    Nyström, Kaj
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Polidoro, Sergio
    A Note on Harnack Inequalities and Propagation Sets for a Class of Hypoelliptic Operators2010Inngår i: Potential Analysis, ISSN 0926-2601, E-ISSN 1572-929X, Vol. 33, nr 4, s. 341-354Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    In this paper we are concerned with Harnack inequalities for non-negative solutions u:Ω→ℝ to a class of second order hypoelliptic ultraparabolic partial differential equations in the form Lu:=∑j=1mX2ju+X0u−∂tu=0 where Ω is any open subset of ℝN + 1, and the vector fields X1, ..., Xm and X0t are invariant with respect to a suitable homogeneous Lie group. Our main goal is the following result: for any fixed (x0, t0) ∈ Ω we give a geometric sufficient condition on the compact sets K⊆Ω for which the Harnack inequality supK u≤CK u(x0, t0) holds for all non-negative solutions u to the equation Lu=0. We also compare our result with an abstract Harnack inequality from potential theory.

  • 5.
    Frentz, Marie
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Garofalo, Nicola
    Department of Mathematics, Purdue University, West Lafayette IN 47907-1968.
    Götmark, Elin
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Munive, Isidro
    Department of Mathematics, Purdue University, West Lafayette IN 47907-1968.
    Nyström, Kaj
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Non-divergence form parabolic equations associated with non-commuting vector fields: Boundary behavior of nonnegative solutions2012Inngår i: Annali della Scuola Normale Superiore di Pisa (Classe Scienze), Serie V, ISSN 0391-173X, E-ISSN 2036-2145, Vol. 11, nr 2, s. 437-474Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    In a cylinder Omega(T) = Omega x (0, T) subset of R-+(n+1) we study the boundary behavior of nonnegative solutions of second order parabolic equations of the form

    H u = Sigma(m)(i,j=1) a(ij)(x, t)XiX (j)u - partial derivative(t)u = 0, (x, t) is an element of R-+(n+1),

    where X = {X-l, . . . , X-m} is a system of C-infinity vector fields inR(n) satisfying Hormander's rank condition (1.2), and Omega is a non-tangentially accessible domain with respect to the Carnot-Caratheodory distance d induced by X. Concerning the matrix-valued function A = {a(ij)}, we assume that it is real, symmetric and uniformly positive definite. Furthermore, we suppose that its entries a(ij) are Holder continuous with respect to the parabolic distance associated with d. Our main results are: I) a backward Harnack inequality for nonnegative solutions vanishing on the lateral boundary (Theorem 1.1); 2) the Holder continuity up to the boundary of the quotient of two nonnegative solutions which vanish continuously on a portion of the lateral boundary (Theorem 1.2); 3) the doubling property for the parabolic measure associated with the operator H (Theorem 1.3). These results generalize to the subelliptic setting of the present paper, those in Lipschitz cylinders by Fabes, Safonov and Yuan in [20, 39]. With one proviso: in those papers the authors assume that the coefficients a(ij) be only bounded and measurable, whereas we assume Holder continuity with respect to the intrinsic parabolic distance.

  • 6.
    Frentz, Marie
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Götmark, Elin
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Nyström, Kaj
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    The obstacle problem for parabolic non-divergence form operators of Hörmander type2012Inngår i: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 252, nr 9, s. 5002-2041Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    In this paper we establish the existence and uniqueness of strong solutions to the obstacle problem for a class of parabolic sub-elliptic operators in non-divergence form structured on a set of smooth vector fields in Rn, X={X1,…,Xq}X={X1,…,Xq}, q⩽n, satisfying Hörmanderʼs finite rank condition. We furthermore prove that any strong solution belongs to a suitable class of Hölder continuous functions. As part of our argument, and this is of independent interest, we prove a Sobolev type embedding theorem, as well as certain a priori interior estimates, valid in the context of Sobolev spaces defined in terms of the system of vector fields.

  • 7.
    Frentz, Marie
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Nyström, Kaj
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Adaptive stochastic weak approximation of degenerate parabolic equations of Kolmogorov type2010Inngår i: Journal of Computational and Applied Mathematics, ISSN 0377-0427, E-ISSN 1879-1778, Vol. 234, nr 1, s. 146-164Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    Degenerate parabolic equations of Kolmogorov type occur in many areas of analysis and applied mathematics. In their simplest form these equations were introduced by Kolmogorov in 1934 to describe the probability density of the positions and velocities of particles but the equations are also used as prototypes for evolution equations arising in the kinetic theory of gases. More recently equations of Kolmogorov type have also turned out to be relevant in option pricing in the setting of certain models for stochastic volatility and in the pricing of Asian options. The purpose of this paper is to numerically solve the Cauchy problem, for a general class of second order degenerate parabolic differential operators of Kolmogorov type with variable coefficients, using a posteriori error estimates and an algorithm for adaptive weak approximation of stochastic differential equations. Furthermore, we show how to apply these results in the context of mathematical finance and option pricing. The approach outlined in this paper circumvents many of the problems confronted by any deterministic approach based on, for example, a finite-difference discretization of the partial differential equation in itself. These problems are caused by the fact that the natural setting for degenerate parabolic differential operators of Kolmogorov type is that of a Lie group much more involved than the standard Euclidean Lie group of translations, the latter being relevant in the case of uniformly elliptic parabolic operators.

  • 8.
    Frentz, Marie
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Nyström, Kaj
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Pascucci, Andrea
    Polidoro, Sergio
    Optimal regularity in the obstacle problem for Kolmogorov operators related to American Asian options2010Inngår i: Mathematische Annalen, ISSN 0025-5831, E-ISSN 1432-1807, Vol. 347, nr 4, s. 805-838Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    In this paper we prove optimal interior regularity for solutions to the obstacle problem for a class of second order differential operators of Kolmogorov type. We treat smooth obstacles as well as non-smooth obstacles. All our proofs follow the same line of thought and are based on blow-ups, compactness, barriers and arguments by contradiction. The problem considered arises in financial mathematics, when considering path-dependent derivative contracts with early exercise feature.

  • 9.
    Götmark, Elin
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Nyström, Kaj
    Boundary behavior of non-negative solutions to degenerate sub-elliptic equations2013Inngår i: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 254, nr 8, s. 3431-3460Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    Let X = {X-1, ..., X-m} be a system of C-infinity vector fields in R-n satisfying Hormander's finite rank condition and let Omega be a non-tangentially accessible domain with respect to the Carnot-Caratheodory distance d induced by X. We study the boundary behavior of non-negative solutions to the equation Lu = Sigma(i, j -1) X-i*(a(ij)X(j)u) = Sigma X-i, j=1(i)*(x)(aij(x)X-j(x)u(x)) = 0 for some constant beta >= 1 and for some non-negative and real-valued function lambda = lambda(x). Concerning kappa we assume that lambda defines an A(2)-weight with respect to the metric introduced by the system of vector fields X =, {X-1,..., X-m}. Our main results include a proof of the doubling property of the associated elliptic measure and the Holder continuity up to the boundary of quotients of non-negative solutions which vanish continuously on a portion of the boundary. Our results generalize previous results of Fabes et al. (1982, 1983) [18-20] (m = n, {X-(1), ..., X-m} = {partial derivative(x1), ...., partial derivative x(n)}, A is an A(2)-weight) and Capogna and Garofalo (1998) [6] (X = {X-1,..., X-m} satisfies Hormander's finite rank condition and X(x) equivalent to lambda A for some constant lambda). One motivation for this study is the ambition to generalize, as far as possible, the results in Lewis and Nystrom (2007, 2010, 2008) [35-38], Lewis et al. (2008) [34] concerning the boundary behavior of non-negative solutions to (Euclidean) quasi-linear equations of p-Laplace type, to non-negative solutions, to certain sub-elliptic quasi-linear equations of p-Laplace type. (C) 2013 Elsevier Inc. All rights reserved.

  • 10. Lewis, John L.
    et al.
    Lundström, Niklas
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Nyström, Kaj
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Boundary Harnack inequalities for operators of p-Laplace type in Reifenberg flat domains2008Inngår i: Perspectives in Partial Differential Equations, Harmonic Analysis and Applications: A Volume in Honor of Vladimir G. Maz'ya's 70th Birthday / [ed] Dorina Mitrea and Marius Mitrea, American Mathematical Society (AMS), 2008, Vol. 79, s. 229-266Kapittel i bok, del av antologi (Fagfellevurdert)
    Abstract [en]

    In this paper we highlight a set of techniques that recently have been used to establish boundary Harnack inequalities for p-harmonic functions vanishing on a portion of the boundary of a domain which is ‘flat’ in the sense that its boundary is well-approximated by hyperplanes. Moreover, we use these techniques to establish new results concerning boundary Harnack inequalities and the Martin boundary problem for operators of p-Laplace type with variable coefficients in Reifenberg flat domains.

  • 11. Lewis, John L
    et al.
    Nyström, Kaj
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    New results for p harmonic functions2011Inngår i: Pure and Applied Mathematics Quarterly, ISSN 1558-8599, E-ISSN 1558-8602, Vol. 7, nr 2, s. 345-363Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    In this paper we first discuss new results of the authors concerning a boundary Harnack inequality and Holder continuity up to the boundary for the ratio of two positive pharmonic functions, 1 < p < infinity, which vanish on a portion of a Lipschitz domain. Second we discuss applications of these results to the Martin boundary problem for p harmonic functions and to certain boundary regularity-free boundary problems.

  • 12. Lewis, John L.
    et al.
    Nyström, Kaj
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Regularity of flat free boundaries in two-phase problems for the p-Laplace operator2012Inngår i: Annales de l'Institut Henri Poincare. Analyse non linéar, ISSN 0294-1449, E-ISSN 1873-1430, Vol. 29, nr 1, s. 83-108Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    In this paper we continue the study in Lewis and Nystrom (2010) [19], concerning the regularity of the free boundary in a general two-phase free boundary problem for the p-Laplace operator, by proving regularity of the free boundary assuming that the free boundary is close to a Lipschitz graph.

  • 13. Lewis, John L
    et al.
    Nyström, Kaj
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Regularity of Lipschitz free boundaries in two-phase problems for the p-Laplace operator2010Inngår i: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 225, nr 5, s. 2565-2597Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    In this paper we study the regularity of the free boundary in a general two-phase free boundary problem for the p-Laplace operator and we prove, in particular, that Lipschitz free boundaries are C(1,gamma)-smooth for some gamma is an element of (0, 1). As part of our argument, and which is of independent interest, we establish a Hopf boundary type principle for non-negative p-harmonic functions vanishing on a portion of the boundary of a Lipschitz domain.

  • 14. Lewis, John L.
    et al.
    Nyström, Kaj
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Poggi-Corradini, Pietro
    p Harmonic Measure in Simply Connected Domains2011Inngår i: Annales de l'Institut Fourier, ISSN 0373-0956, E-ISSN 1777-5310, Vol. 61, nr 2, s. 689-715Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    Let Omega be a bounded simply connected domain in the complex plane, C. Let N be a neighborhood of partial derivative Omega, let p be fixed, 1 < p < infinity, and let (u) over cap be a positive weak solution to the p Laplace equation in Omega boolean AND N. Assume that (u) over cap has zero boundary values on partial derivative Omega in the Sobolev sense and extend (u) over cap to N \ Omega by putting 11 E 0 on N Then there exists a positive finite Borel measure (mu) over cap on C with support contained in partial derivative Omega and such that integral vertical bar del(u) over cap vertical bar(p-2) <del(u) over cap, del phi > dA = - integral phi d (mu) over cap whenever phi is an element of C(0)(infinity)(N). If p = 2 and if (u) over cap is the Green function for Omega with pole at x is an element of Omega\(N) over bar then the measure (mu) over cap coincides with harmonic measure at x, omega = omega(x), associated to the Laplace equation. In this paper we continue the studies initiated by the first author by establishing new results, in simply connected domains, concerning the Hausdorff dimension of the support of the measure (mu) over cap. In particular, we prove results, for 1 < p < infinity, p not equal 2, reminiscent of the famous result of Makarov concerning the Hausdorff dimension of the support of harmonic measure in simply connected domains.

  • 15. Lewis, John
    et al.
    Nyström, Kaj
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Matematiska institutionen.
    Boundary behavior and the Martin boundary problem for p harmonic functions in Lipschitz domains2010Inngår i: Annals of Mathematics, ISSN 0003-486X, E-ISSN 1939-8980, Vol. 172, nr 3, s. 1907-1948Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    In a previous article, we proved a boundary Harnack inequality for the ratio of two positive p harmonic functions, vanishing on a portion of the boundary of a Lipschitz domain. In the current paper we continue our study by showing that this ratio is Holder continuous up to the boundary. We also consider the Martin boundary of certain domains and the corresponding question of when a minimal positive p harmonic function (with respect to a given boundary point w) is unique up to constant multiples. In particular we show that the Martin boundary can be identified with the topological boundary in domains that are convex or C(1). Minimal positive p harmonic functions relative to a boundary point w in a Lipschitz domain are shown to be unique, up to constant multiples, provided the boundary is sufficiently flat at w.

  • 16.
    Lundström, Niklas L P
    et al.
    Department of Mathematics, Uppsala University, Sweden.
    Nyström, Kaj
    Department of Mathematics, Uppsala University, Sweden.
    Olofsson, Marcus
    Department of Mathematics, Uppsala University, Sweden.
    Systems of variational inequalities in the context of optimal switching problems and operators of Kolmogorov type2014Inngår i: Annali di Matematica Pura ed Applicata, ISSN 0373-3114, E-ISSN 1618-1891, Vol. 193, nr 4, s. 1213-1247Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    In this paper we study the system where . A special case of this type of system of variational inequalities with terminal data occurs in the context of optimal switching problems. We establish a general comparison principle for viscosity sub- and supersolutions to the system under mild regularity, growth, and structural assumptions on the data, i.e., on the operator and on continuous functions , , and . A key aspect is that we make no sign assumption on the switching costs and that is allowed to depend on as well as . Using the comparison principle, the existence of a unique viscosity solution to the system is constructed as the limit of an increasing sequence of solutions to associated obstacle problems. Having settled the existence and uniqueness, we subsequently focus on regularity of beyond continuity. In this context, in particular, we assume that belongs to a class of second-order differential operators of Kolmogorov type of the form: where . The matrix is assumed to be symmetric and uniformly positive definite in . In particular, uniform ellipticity is only assumed in the first coordinate directions, and hence, may be degenerate.

  • 17.
    Lundström, Niklas L.P.
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Nyström, Kaj
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    On a two-phase free boundary condition for p-harmonic measures2009Inngår i: Manuscripta mathematica, ISSN 0025-2611, E-ISSN 1432-1785, Vol. 129, nr 2, s. 231-249Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    Let Ωi⊂Rn,i∈{1,2} , be two (δ, r 0)-Reifenberg flat domains, for some 0<δ<δ^ and r 0 > 0, assume Ω1∩Ω2=∅ and that, for some w∈Rn and some 0 < r, w∈∂Ω1∩∂Ω2,∂Ω1∩B(w,2r)=∂Ω2∩B(w,2r) . Let p, 1 < p < ∞, be given and let u i , i∈{1,2} , denote a non-negative p-harmonic function in Ω i , assume that u i , i∈{1,2}, is continuous in Ω¯i∩B(w,2r) and that u i = 0 on ∂Ωi∩B(w,2r) . Extend u i to B(w, 2r) by defining ui≡0 on B(w,2r)∖Ωi. Then there exists a unique finite positive Borel measure μ i , i∈{1,2} , on R n , with support in ∂Ωi∩B(w,2r) , such that if ϕ∈C∞0(B(w,2r)) , then∫Rn|∇ui|p−2⟨∇ui,∇ϕ⟩dx=−∫Rnϕdμi.Let Δ(w,2r)=∂Ω1∩B(w,2r)=∂Ω2∩B(w,2r) . The main result proved in this paper is the following. Assume that μ 2 is absolutely continuous with respect to μ 1 on Δ(w, 2r), d μ 2 = kd μ 1 for μ 1-almost every point in Δ(w, 2r) and that logk∈VMO(Δ(w,r),μ1) . Then there exists δ~=δ~(p,n)>0 , δ~<δ^ , such that if δ≤δ~ , then Δ(w, r/2) is Reifenberg flat with vanishing constant. Moreover, the special case p = 2, i.e., the linear case and the corresponding problem for harmonic measures, has previously been studied in Kenig and Toro (J Reine Angew Math 596:1–44, 2006).

  • 18.
    Lundström, Niklas L.P.
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Nyström, Kaj
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    The Boundary Harnack Inequality for Solutions to Equations of Aronsson type in the Plane2011Inngår i: Annales Academiae Scientiarum Fennicae Mathematica, ISSN 1239-629X, E-ISSN 1798-2383, Vol. 36, s. 261-278Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    In this paper we prove a boundary Harnack inequality for positive functions which vanish continuously on a portion of the boundary of a bounded domain \Omega \subset R2 and which are solutions to a general equation of p-Laplace type, 1 < p < \infty. We also establish the same type of result for solutions to the Aronsson type equation \nabla (F(x,\nabla u)) \cdot F\eta(x,\nabla u) = 0. Concerning \Omega we only assume that \partial\Omega is a quasicircle. In particular, our results generalize the boundary Harnack inequalities in [BL] and [LN2] to operators with variable coefficients.

  • 19.
    Lundström, Niklas
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Nyström, Kaj
    Uppsala Univ, Dept Math.
    Olofsson, Marcus
    Uppsala Univ, Dept Math.
    Systems of variational inequalities for non-local operators related to optimal switching problems: existence and uniqueness2014Inngår i: Manuscripta mathematica, ISSN 0025-2611, E-ISSN 1432-1785, Vol. 145, nr 3-4, s. 407-432Artikkel i tidsskrift (Fagfellevurdert)
  • 20.
    Nyström, Kaj
    Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
    A framework for the valuation and risk management of non-maturing liabilities2005Rapport (Annet vitenskapelig)
  • 21.
    Nyström, Kaj
    Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
    An estimate of a polynomial capacity1998Inngår i: Potential Analysis, ISSN 0926-2601, Vol. 9, nr 3, s. 217-227Artikkel i tidsskrift (Fagfellevurdert)
  • 22.
    Nyström, Kaj
    Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
    Boundary value problems and duality between Lp Dirichlet and regularity problems for second order parabolic systems in non-cylindrical domains2006Inngår i: Collectanea Mathematica, ISSN 0010-0757, Vol. 57, nr 1, s. 93-119Artikkel i tidsskrift (Fagfellevurdert)
  • 23.
    Nyström, Kaj
    Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
    Boundary value problems for parabolic Lamé systems in time-varying domains1999Inngår i: Indiana University Mathematics Journal, ISSN 0022-2518, Vol. 48, nr 4, s. 1285-1355Artikkel i tidsskrift (Fagfellevurdert)
  • 24.
    Nyström, Kaj
    Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
    Caloric measure and Reifenberg flatness2006Inngår i: Annales Academiae Scientiarium Fennicae. Mathematica, ISSN 1239-629X, Vol. 31, s. 405-436Artikkel i tidsskrift (Fagfellevurdert)
  • 25.
    Nyström, Kaj
    Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
    Free boundary regularity of multi-dimensional American options through blow-ups and global solutions2007Inngår i: Journal of Computational Mathematics and Optimization, Vol. 3, nr 1, s. 39-76Artikkel i tidsskrift (Fagfellevurdert)
  • 26.
    Nyström, Kaj
    Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
    Harmonic analysís, quadratic forms and asymptotic expansions of risk measures2008Inngår i: Applied Mathematical Sciences, Vol. 2, nr 21-24, s. 1023-1052Artikkel i tidsskrift (Fagfellevurdert)
  • 27.
    Nyström, Kaj
    Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
    Integrability of Green potentials in fractal domains1996Inngår i: Arkiv för Matematik, ISSN 0004-2080, Vol. 34, s. 335-381Artikkel i tidsskrift (Fagfellevurdert)
  • 28.
    Nyström, Kaj
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    On an inverse type problem for the heat equation in parabolic regular graph domains2012Inngår i: Mathematische Zeitschrift, ISSN 0025-5874, E-ISSN 1432-1823, Vol. 270, nr 1-2, s. 197-222Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    In this paper we prove some results concerning inverse/free boundary type problems, below the continuous threshold, for the heat equation in the setting of parabolic regular graph domains.

  • 29.
    Nyström, Kaj
    Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
    On area integral estimates for solutions to parabolic systems in time-varying and non-smooth cylinders2008Inngår i: Transactions of the American Mathematical Society, ISSN 0002-9947, Vol. 360, nr 6, s. 2987-3017Artikkel i tidsskrift (Fagfellevurdert)
  • 30.
    Nyström, Kaj
    Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
    On blow-ups and the classification of global solutions to parabolic free boundary problems2005Rapport (Annet vitenskapelig)
  • 31.
    Nyström, Kaj
    Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
    On blow-ups and the classification of global solutions to parabolic free boundary problems2006Inngår i: Indiana University Mathematics Journal, ISSN 0022-2518, Vol. 55, nr 4, s. 1233-1290Artikkel i tidsskrift (Fagfellevurdert)
  • 32.
    Nyström, Kaj
    Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
    On deposit volumes and the valuation of non-maturing liabilities2008Inngår i: Journal of Economic Dynamics & Control, ISSN 0165-1889, Vol. 32, nr 3, s. 709-756Artikkel i tidsskrift (Fagfellevurdert)
  • 33.
    Nyström, Kaj
    Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
    On the behaviour near expiration for multidimensional American options2006Rapport (Annet vitenskapelig)
  • 34.
    Nyström, Kaj
    Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
    On the behaviour near expiry for multi-dimensional American options2008Inngår i: Journal of Mathmatical Analysis and Applications, ISSN 0022-247X, Vol. 339, nr 1, s. 644-654Artikkel i tidsskrift (Fagfellevurdert)
  • 35.
    Nyström, Kaj
    Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
    On the rating and pricing of mortgage portfolios through structured finance2007Inngår i: Journal of Risk Model Validation, Vol. 1, nr 4, s. 1-61Artikkel i tidsskrift (Fagfellevurdert)
  • 36.
    Nyström, Kaj
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Regularity below the continuous threshold in a two-phase parabolic free boundary problem2006Inngår i: Mathematica Scandinavica, ISSN 0025-5521, E-ISSN 1903-1807, Vol. 99, nr 2, s. 257-286Artikkel i tidsskrift (Fagfellevurdert)
  • 37.
    Nyström, Kaj
    Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
    Smoothness properties of solutions to Dirichlet problems in domains with a fractal boundary1994Doktoravhandling, monografi (Annet vitenskapelig)
  • 38.
    Nyström, Kaj
    Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
    Structuring, credit quality and pricing of cash flow CDOS2006Rapport (Annet vitenskapelig)
  • 39.
    Nyström, Kaj
    Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
    The Dirichlet problem for second order parabolic operators1997Inngår i: Indiana University Mathematics Journal, ISSN 0022-2518, Vol. 46, nr 1, s. 183-245Artikkel i tidsskrift (Fagfellevurdert)
  • 40.
    Nyström, Kaj
    et al.
    Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
    Hoffman, S.
    Dirichlet problems for a non-stationary linearized system of Navier-Stokes equations in non-cylindrical domains.2002Inngår i: Journal of Methods and Applications of Analysis, Vol. 9, nr 1, s. 13-98Artikkel i tidsskrift (Fagfellevurdert)
  • 41.
    Nyström, Kaj
    et al.
    Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
    Hoffman, S.
    Lewis, J.
    Caloric measure in parabolic flat domains2004Inngår i: Duke Mathematical Journal, ISSN 0012-7094, Vol. 122, nr 2, s. 281-346Artikkel i tidsskrift (Fagfellevurdert)
  • 42.
    Nyström, Kaj
    et al.
    Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
    Hoffman, S.
    Lewis, J.
    Existence of big pieces of graphs for parabolic problems2003Inngår i: Annales Academiae Scientiarium Fennicae. Mathematica, ISSN 1239-629X, Vol. 28, s. 355-384Artikkel i tidsskrift (Fagfellevurdert)
  • 43.
    Nyström, Kaj
    et al.
    Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
    Kukavica, Igor
    Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
    Unique continuation on the boundary for Dini domains1998Inngår i: Proceedings of the American Mathematical Society, ISSN 0002-9939, Vol. 126, nr 2, s. 441-446Artikkel i tidsskrift (Fagfellevurdert)
  • 44.
    Nyström, Kaj
    et al.
    Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
    Lewis, J.
    Boundary behaviour for p-harmonic functions in Lipschitz and starlike Lipschitz ring domains2007Inngår i: Annales Scientifigues de L'Ecole Normale Superieure, ISSN 0012-9593, Vol. 40, nr 5, s. 765-813Artikkel i tidsskrift (Fagfellevurdert)
  • 45.
    Nyström, Kaj
    et al.
    Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
    Lewis, J.
    Boundary behaviour of p-harmonic functions in domains beyond Lipschitz domains2008Inngår i: Advances in the Calculus of Variations, Vol. 1, s. 1-38Artikkel i tidsskrift (Fagfellevurdert)
  • 46.
    Nyström, Kaj
    et al.
    Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
    Lewis, J.
    On a parabolic symmetry problem2007Inngår i: Revista Matemática Iberoamericana, ISSN 0213-2230, Vol. 23, nr 2, s. 513-536Artikkel i tidsskrift (Fagfellevurdert)
  • 47.
    Nyström, Kaj
    et al.
    Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
    Lewis, J.
    Regularity and free boundary regularity for the p-Laplacian in Lipschitz and C1-domains2008Inngår i: Annales Academiae Scientiarium Fennicae, ISSN 1239-629X, Vol. 33, s. 1-26Artikkel i tidsskrift (Fagfellevurdert)
  • 48.
    Nyström, Kaj
    et al.
    Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
    Lewis, J.
    The boundary Harnack inequality for infinity harmonic functions in the plane2008Inngår i: Proceedings of the American Mathematical Society, ISSN 0002-9939, Vol. 136, s. 1311-1323Artikkel i tidsskrift (Fagfellevurdert)
  • 49.
    Nyström, Kaj
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Pascucci, Andrea
    Polidoro, Sergio
    Regularity near the initial state in the obstacle problem for a class of hypoelliptic ultraparabolic operators2010Inngår i: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 249, nr 8, s. 2044-2060Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    This paper is devoted to a proof of regularity, near the initial state, for solutions to the Cauchy-Dirichlet and obstacle problem for a class of second order differential operators of Kolmogorov type. The approach used here is general enough to allow us to consider smooth obstacles as well as non-smooth obstacles.

  • 50.
    Nyström, Kaj
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Skoglund, Jimmy
    A credit risk model for large dimensional portfolios with application to economic capital2005Rapport (Annet vitenskapelig)
12 1 - 50 of 60
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